Quantization of the Gravitational Field in the Expanding Universe

概要

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An attempt is made to quantize, in the linear approximation, the gravitational field in the expanding universe, from the standpoint of our canonical formalism worked out in a previous paper. By using a transformation function and decomposing suitably the field operator into positive and negative frequency parts, explicit forms of the four-dimensional commutation function and the propagation function are derived in the case of three special world models. There appears the difficulty of divergence term in the propagation function, with the exception of one model. Our propagation function, however, satisfies the relation between Hadamard's elementary solution and the propagation function postulated by De Witt and Brehme. In addition, a connection between the propagation function and the time-ordered product of field operators is established by introducing a state of quasi-vacuum in parallel with our decomposition mentioned above. Finally, detailed discussions of several difficulties are given in connection with the problem of causality. It is suggested from our analysis that the quantum theory and the general relativity cannot rigorously be compatible, so far as we stand on the hypothesis of the expanding universe.
理論物理学刊行会の論文
1963-02-25